Solve y-x^-1/x-y^-1*x-y^-1/x^-2+{y^-2}

$$\frac{ y- { x }^{ -1 } }{ x- { y }^{ -1 } } \times \frac{ x- { y }^{ -1 } }{ { x }^{ -2 } + { y }^{ -2 } }$$

Evaluate

$$\frac{xy^{2}\left(xy-1\right)}{x^{2}+y^{2}}$$

Short Solution Steps

Multiply \frac{y-x^{-1}}{x-y^{-1}} times \frac{x-y^{-1}}{x^{-2}+y^{-2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(y-x^{-1}\right)\left(x-y^{-1}\right)}{\left(x-y^{-1}\right)\left(x^{-2}+y^{-2}\right)}$$

Cancel out x-\frac{1}{y} in both numerator and denominator.

$$\frac{y-\frac{1}{x}}{x^{-2}+y^{-2}}$$

Expand

$$\frac{x^{2}y^{3}-xy^{2}}{x^{2}+y^{2}}$$

Short Solution Steps

Multiply \frac{y-x^{-1}}{x-y^{-1}} times \frac{x-y^{-1}}{x^{-2}+y^{-2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(y-x^{-1}\right)\left(x-y^{-1}\right)}{\left(x-y^{-1}\right)\left(x^{-2}+y^{-2}\right)}$$

Cancel out x-\frac{1}{y} in both numerator and denominator.

$$\frac{y-\frac{1}{x}}{x^{-2}+y^{-2}}$$